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Nonlinear Control (FRTN05)

Computer Exercise 4

Last updated: Spring of 2011

Introduction

Goal

The goal of the computer exercise is to simulate parts of a JAS 39 Gripen (amilitary aircraft)control system, and to use the describing function methodto analyze Pilot-Induced Oscillation(PIO).

Contents

The exercise is performed in Matlab and Simulink, either at the depart-

ment or at any other computer that has Matlab with the Control System

Toolbox, and Simulink. You will need some files for the exercise. These

are available at the course homepage. Copy the files into the directory

where you will run. To find out if you have the control system toolbox

writehelp control.(We will use the commands ss, tf, bode, nyquist,evalfr. Use help to find out how they work.)

The first part, Section 1, contains small examples of how Simulink

works. The following part of the exercise can be hard, so do not spend

too much time on the introduction to Simulink! The Simulink introduc-tion can be skipped if you already are familiar with Simulink. Section 2

contains the main problem; to analyze PIO of JAS.

1. Introduction to Simulink

Simulink is a simulation program based upon Matlab. There are sev-

eral ways to define a model. One can work graphically and connect block-

diagrams with predefined blocks. Alternatively one can give the mathemat-

ical description in forms of differential equations in an m-file (the format forprograms written in the Matlab programming language). Matlab/Simulinksupports both these representations as well as combinations. Furthermore

one can use descriptions that include a hierarchy of connected subsystems.

To understand how models are described and simulated using block

diagrams, it is best to run small examples on a computer. The rest of

Section 1 shows some examples. If you are familiar with Simulink you can

go directly to Section 2.

1.1 How to Start Simulink

Start Matlab. Then give the command simulink in Matlab. This gives a

window with blocks as in Figure 1. Each block represents a library that

contains several building blocks.

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Simulink Block Library 3.0

Copyright (c) 19901998 by The MathWorks, Inc.

Sources Sinks

In1 Out1

Signals& Systems

NonlinearMath Functions& Tables

DiscreteContinuous

Figure 1 Available Simulink block diagram libraries

1.2 A Simple System

Click onFilein the Simulink-window and chooseNew->Model. Click on the

library Continuous and move a Transfer Fcn to the new window called

Untitled. Do the same with Sources->Step Fcnand Sinks->Scope. Draw

arrows (left mouse button) and connect the ports on the block. You should

now have a block diagram as in Figure 2.

1

s+1

Transfer FcnStep Scope

Figure 2 A simple Simulink system

Choose Simulation->Parameters in the window called Untitled. Set

Stop timeto 5. Open a window for the Scopeby double clicking on it. Start

a simulation by Simulation->Start (or by pressing Ctrl-t in the windowcalled Untitled).

How to Change a System To change the system to

1

s2 + 0.5s+ 2

you double-click on the block Transfer Fcn and change Denominator to

[1 0.5 2]. Simulate the new system(Simulation->Start or Ctrl-t). Changeparameters in theSimulationmenu and the scales in the block Scopeuntil

you are satisfied.

How to Change an Input Signal To change the input signal, start

with removing the blockStep Fcnby clicking on it and delete it by pressing

Delete (or usingEdit->Cutor pressing Ctrl-x). Replace it by aSources->SignalGeneratorblock. Double-click onSignal Generatorand select a wave form,

amplitude and frequency. Also changeSimulation->Parameters->Stop Time

to 99999 and press Simulation->Start. This gives an infinite simulation

that can be stopped by pressing Simulation->Stop (or Ctrl-t). Can theamplitude of the input signal be changed during simulation? Also try to

change the block Transfer Fcnduring simulation.

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How to Use Matlab Variables in Blocks Note that variables defined

in the Matlab environment can be used in Simulink. Define numerator and

denominator by writing the following in the Matlab window.

num=[1 1]

den=[1 2 3 4]

ChangeTransfer Fcn->NumeratortonumandTransfer Fcn->Denominator

to den.

How to Save Results to Matlab Variables To save input and output,

move two copies of the block Sinks->To Workspace. Connect these with

the input and output to the block Transfer Fcn. Get a Sources->Clock

and connect it to a Sinks->To Workspace. Double click on the Workspace

blocks to be able to change the variable names to u,y, and t respec-

tively. Also change Save format to the value Array. The window should

look something like Fig. 3.

Figure 3 How to save results to Matlab variables in Simulink

How to Use Simulation Results in Matlab Calculations Let the

input signal be a sinusoidal with frequency 0.1 rad/s and amplitude 1.Do a simulation that is long enough for the output to become stationary.

Compute

n=length(y)

max(y(n/2:n))and compare this with the theoretical value G(0.1i).

>>g=tf(num,den)

Transfer function:

s + 1

---------------------

s ^ 3 + 2 s ^ 2 + 3 s + 4

>> abs(evalfr(g,0.1*i))

ans =

0.2518

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How to Save Systems Use File-Save Asor File->Save.

1.3 A Flow System

Consider a simple tank as in the basic control course

h = 1A

(uq)

q = a

2h.

This can be implemented in Simulink as in Figure 4. The function f(u)

2h

1

qs

1

Integrator

1/A

Gain

f(u)

Fcn

1

In

Figure 4 A tank system

has the value a*sqrt(2*g*u[1]). The summation block has been given two

inputs with different signs by assigning the string -+ to Sum->List of

Signs. The summation and the Gain blocks are found in the Math library

and the Fcn block is found in the Functions & Tables library. The small

ellipses, that are contained in theSignals & Systemslibrary, tell Simulink

what should be considered inputs and outputs to this (sub)system. The

block titles can be changed by clicking on them. Mark the entire systemby holding the left mouse bottom pressed and drawing a rectangle around

it. Then choose Edit->Create Subsystem. The result is that the system is

represented by one block. Use Edit->Copy to create the following double-

tank system. Use the commandstrimandlinmodto find a linearized model

1

Out1

In

q

h

Subsystem1

In

q

h

Subsystem

1

In1

Figure 5 Two tanks and some connections

of the double tank around h01 = h02 = 0.1. Use the parameters A1 = A2 =

2.7 103, a1 = a2 = 7.0 106, =9.8. Plot the Nyquist curve using the

commandnyquist.

>> A=2.7e-3;a=7e-6;g=9.8;

>> [x0,u0,y0]=trim(flow,[0.1 0.1],[],0.1)

Warning: Output port 2 of block twotank/Subsystem is not connected.

Warning: Output port 1 of block twotank/Subsystem1 is not connected.

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x0 =

0.1000

0.1000

u0 =

9.7995e-06

y0 =0.1000

>> [aa,bb,cc,dd]=linmod(flow,x0,u0);

>> sys=ss(aa,bb,cc,dd);

>> bode(sys)

Alternative: Linearization in Simulink;

By right-clicking on a signal connector in a Simulink model you can add

Linearization points (inputs and/or outputs). Use this for the two watertanks.

Start a Control and Estimation Tool Manager by

Tools -> Control Design ->Linear analysis ..., see Fig. 6. Here you

can set the desired operating points, export linearized model to Workspace(Model-> Export to Workspace) and much more.

Repeat the linearization of the system at the same equilibrium point as

above.

Figure 6 View of "Control and Estimation Tool Manager" in Matlab/Simulink.

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2. Example JAS 39 Gripen

2.1 BackgroundModel

We will study simulation and control of the pitch dynamics of a JAS 39

Gripen. The dynamics of an airplane is highly nonlinear. The control sys-tem uses gain scheduling on speed and altitude to compensate for parts

of the nonlinearities. Linear models have been obtained for approximately

50 different operating conditions. The models are given in state-space form

and are the result of extensive wind tunnel experiments and calculations

at SAAB. A controller is designed for each linear model, and a switching

mechanism is used to switch between the different controllers. Many of the

parameters in the models vary considerably within normal operation of the

aircraft. Two of the extreme cases are heavily loaded aircraft at low speed

and unloaded aircraft at high speed. The aircraft is also very sensitive

when the speed is close to Mach 1 and the dynamics change from unstable

to stable. The Mach number gives the speed of the aircraft, and is given

by M=v/a, where a is the velocity of sound.We will work with a linear model with five states for normal load at

M = 0.6 and altitude 1 km. The model is taken from a master thesis atChalmers, see [1]. The state space model is defined in the file jasdata.m.Write

>>jasdata

The model of the aircraft is given by

x= Ax+ Bu,

where the state vector x consists of the seven variables

angle of attack

q pitch rate

pitch angle

e elevator angle

s spoiler angle

xe internal elevator state

xs internal spoiler state

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The inputs in u are given by ue, that is the elevator command, and us,

that is the spoiler command.

= +

airplane fixed x-direction

airplane velocity direction

earth frame

The rudder servos are controlled in inner loops that give the rudder

dynamics

= 1

(0.05s+ 1)(0.008s+ 1)u.

This means that each rudder has two states, the rudder angle (e ands)and the internal rudder state(xe and xs).

The plane is controlled by linear state feedback u = Lx. The feedbackloop stabilizes the states q and . The pitch angle is not stabilized. Thecontrol of this mode is left for the pilot (more about this later). The L-vectorhas been designed using linear quadratic control theory. A reduced model is

used in the design, with the fast rudder dynamics neglected (correspondingto the pole in s = 1/0.008). For this reason, the internal rudder statesare not used in the feedback loop. In reality there are no measurements of

rudder anglese and s. These are instead estimated with a Kalman filter.The total control signal is given by

u(t) = Lx(t) + Kfufpilot

(t),

where Kf is a constant which gives the correct closed loop steady state

gain, and ufpilot

is the filtered pilot command signal

ufpilot

= 1

Tfs+ 1upilot

To see the Simulink model of the aircraft write

>>planemodelThis gives the block diagram in Figure 7.

2.2 The Pilot Normal Function

The pilot is modeled as a PD-controller with time delay of 0.3s. The pilot

transfer function is given by

upilot = Kp1 +Tds

1 +Td/Nse0.3s (re f ),

whereKp= 0.2,Td= 0.5, and N=10. Modelling a pilot as a PD-controller

can be motivated by that it is natural that a pilot gives control commands

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command

upilot

t

time

x

states

reference

1

T_f.s+1

prefilter

x = Ax+Buy = Cx+Du

planedynamics

theta

pitch angle

pilot 1

L

Kf

Clock

Ctheta

Figure 7 Simulink model of plane and pilot.

proportional to the control error, but is a little more careful when the pitch

angle is changing rapidly.

Assignment 1:Regard the state space model of the aircraft(given by the

A-, B-, C-, andD-matrices). Find the poles of the system (the eigenvaluesof theA-matrix). Is the open loop system unstable? How many inputs and

outputs does the model have? Go through the interconnections in the A-

matrix to see if you can find sub-systems and how these relate to the

open-loop poles(use the state vector information on p.6).

Choose a nominal design for the state feedback by typing

>>design1

Look at the L-matrix. Are the correct states used in the feedback loop?

What are the eigenvalues of A BL? Explain the pole close to the origin(i.e., which state does it correspond to). Why has this pole placement been

chosen?

Simulate the closed-loop system including the pilot and check that itworks properly. Check that the rudder angles are within 0.5radians(useplot(t,x(:,[4 5]))). Why is it important that the rudder angles are not

too large?

2.3 Pilot Induced Oscillations (PIO)

The pilot model used in Assignment 1 can be seen as a pilot who works

rationally and always does the right thing. In an emergency situation the

pilot may panic, and try to compensate the error with maximal command

signals. Imagine yourself in any balancing act. When you are in control,

you can keep the balance using very small motions, but as soon as you are a

little out of control, your movements tends to be very large and eventuallyyou will fall. This is typical for systems with relatively slow dynamics,

which cannot react quick enough to the control signals. If you start to fall

and try to get back up again, because of the slow dynamics you will use

too much efforts to come up so that once your up again you cannot stop,

but will immediately start falling in the other direction.

The phenomena got considerable attention after the crash in Stockholm

in 1993, and was given the name pilot induced oscillations, PIO. In the lab

we will make a simplified analysis using a relay model of the pilot in PIO

mode. The pilot gives maximal joystick commands based on the sign of.

Assignment 2: A relay pilot can be found in the Simulink pilot model

library. Write

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>>pilotlib

Plot the Nyquist curve of the linear system from upilot to. This can bedone by deleting the feedback path from , connecting input and outputconnections at appropriate places, saving the system to a new file and using

the linmod andnyquist commands. Use help to see what the commands

do. The describing function for a relay is

N(A) = 4D

A,

What is the amplitudeD of the relay pilot? Use describing function anal-

ysis to estimate the amplitude and frequency of a possible limit cycle by

reading out the intersection between 1/N(A) and the nyquist plot. (Youcan easily zoom and mark points on the Nyquist plot to read out e.g., fre-

quencies).

Change pilot in the plane model by deleting the first pilot and instead

choose the relay pilot. Simulate the system. How good is the prediction

of the limit cycle with the simulation results?

Assignment 3:As you can see, the amplitude is relatively moderate. This

is because the flight condition is high speed and high altitude. Let us

anyway discuss two possible ways to reduce PIO:

Use the commanddesign2 to changeL andKfto a faster design. Is

the PIO amplitude decreased?

Use the command design3 to make the filter faster by reducing the

filter constant toTf =0.03. Is the PIO amplitude decreased?

Compare the Nyquist curves from upilot to for the different designs (de-

sign 1-3). In the light of the describing function method; which designreduces PIO amplitude? Can you find some drawback with this method?

Hint: Simulate the system with the normal pilot from assignment 1 using

the different designs.

PhD Assignment: There are no rate limitations on rudders in the model.

Rate limitations were also part of the source of control problems on the

JAS. Introduce rate limitations, for example as in the article by Rundquist

et al., and investigate what happens to the limit cycle. Does it become

unstable? Try to understand the idea of the patented nonlinear filter.

References

[1] Lars Axelsson, Reglerstudier av Back-up regulator fr JAS 39 Gripen,examensarbete CTH/Saab flygdivisionen, 1992.[2] Rundquist et al., Rate Limiters with Phase Compensation in JAS 39Gripen, European Control Conference 1997.

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